Evaluate the following limits. limx→∞(3⋅21−x+x2⋅21−x)

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Evaluate limit: limx→∞(3⋅21−x+x2⋅21−x) limx→a[f(x)+−g(x)]=limx→af(x)+−limx→ag(x) limx→∞(3⋅21−x+x2⋅21−x)=limx→∞(3⋅21−x)+limx→∞(x2⋅21−x) =3⋅limx→∞(21−x)+2⋅limx→∞(x2⋅2−x) =3⋅limx→∞(21−x)+2⋅limx→∞(x2⋅12x) Take 3⋅limx→∞(21−x) Apply exponent rule ax=eln⁡ax=ex⋅ln⁡a =3⋅limx→∞(e(1−x)⋅ln⁡2) =3⋅limx→∞21−x =3⋅limx→∞ex⋅ln⁡2 Apple the Limit Chain Rule: g(x)=xln⁡(2),f(u)=eu =−∞⋅ln⁡2 =−∞ 3⋅limx→∞ex⋅ln⁡2=3⋅e−∞ =3⋅0 =0 Take 2⋅limx→∞x2⋅12x if sum an converges,then limn→∞(an)=0 Apply ratio test and check weather series is convergent of divergent if |an+1an|≤q eventually for some 0, then ∑n=1x|an| converges, if |an+1an|&amp;gt;1 eventually then ∑n=1xan diverges limx→∞(x22x)=0 =2⋅0 =0 limx→∞(3⋅21−x+x2⋅21−x)=3⋅limx→∞(21−x)+2⋅limx→∞(x2⋅12x) =0+0 =0 Result: 3⋅limx→∞(21−x)+2⋅limx→∞(x2⋅12x)=0

Evaluate the following limits. lim_{xrightarrowinfty}(3cdot2^{1-x}+x^2cdot2^{1-x})

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